Math

Question Find the areas and volumes of a round 11-inch and 8-inch by 10-inch rectangular pan. Determine which has the larger volume.
a) Area of round pan: π(11/2)2\pi (11/2)^2 Area of rectangular pan: 8×108 \times 10 b) Volume of round pan: π(11/2)2×2\pi (11/2)^2 \times 2 Volume of rectangular pan: 8×10×28 \times 10 \times 2 c) Compare the volumes to determine the larger pan.

Studdy Solution

STEP 1

Assumptions
1. The diameter of the round pan is 11 inches.
2. The dimensions of the rectangular pan are 8 inches by 10 inches.
3. Both pans are 2 inches deep.
4. The value of π\pi is needed for calculations involving the round pan.
5. We are comparing the areas of the bases and the volumes of the two pans.

STEP 2

To determine the area of the base of the round pan, we will use the formula for the area of a circle, which is A=πr2A = \pi r^2, where rr is the radius of the circle.

STEP 3

First, we find the radius of the round pan by dividing the diameter by 2.
Radius=Diameter2=11in2Radius = \frac{Diameter}{2} = \frac{11\,in}{2}

STEP 4

Calculate the radius of the round pan.
Radius=5.5inRadius = 5.5\,in

STEP 5

Now, plug in the radius into the area formula for the round pan.
Arearoundpan=π×(5.5in)2Area_{round\,pan} = \pi \times (5.5\,in)^2

STEP 6

Calculate the area of the base of the round pan.
Arearoundpan=π×5.52in2=π×30.25in2Area_{round\,pan} = \pi \times 5.5^2\,in^2 = \pi \times 30.25\,in^2

STEP 7

Use the value of π\pi from the calculator to find the numerical value of the area of the round pan.
Arearoundpan3.14159×30.25in295.0331in2Area_{round\,pan} \approx 3.14159 \times 30.25\,in^2 \approx 95.0331\,in^2

STEP 8

To determine the area of the base of the rectangular pan, we will use the formula for the area of a rectangle, which is A=length×widthA = length \times width.

STEP 9

Plug in the dimensions of the rectangular pan into the area formula.
Arearectangularpan=8in×10inArea_{rectangular\,pan} = 8\,in \times 10\,in

STEP 10

Calculate the area of the base of the rectangular pan.
Arearectangularpan=80in2Area_{rectangular\,pan} = 80\,in^2

STEP 11

Now that we have the areas of both pans, we can proceed to calculate their volumes. The volume of a pan is calculated by multiplying the area of the base by the depth of the pan.

STEP 12

First, calculate the volume of the round pan using its base area and the depth.
Volumeroundpan=Arearoundpan×DepthVolume_{round\,pan} = Area_{round\,pan} \times Depth

STEP 13

Plug in the values for the area of the round pan and the depth.
Volumeroundpan95.0331in2×2inVolume_{round\,pan} \approx 95.0331\,in^2 \times 2\,in

STEP 14

Calculate the volume of the round pan.
Volumeroundpan95.0331in2×2in190.0662in3Volume_{round\,pan} \approx 95.0331\,in^2 \times 2\,in \approx 190.0662\,in^3

STEP 15

Next, calculate the volume of the rectangular pan using its base area and the depth.
Volumerectangularpan=Arearectangularpan×DepthVolume_{rectangular\,pan} = Area_{rectangular\,pan} \times Depth

STEP 16

Plug in the values for the area of the rectangular pan and the depth.
Volumerectangularpan=80in2×2inVolume_{rectangular\,pan} = 80\,in^2 \times 2\,in

STEP 17

Calculate the volume of the rectangular pan.
Volumerectangularpan=80in2×2in=160in3Volume_{rectangular\,pan} = 80\,in^2 \times 2\,in = 160\,in^3

STEP 18

Now we compare the volumes of both pans to determine which one has the larger volume.

STEP 19

Since 190.0662in3>160in3190.0662\,in^3 > 160\,in^3, the round pan has a larger volume than the rectangular pan.
The round pan has the larger volume.

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